Subgroups — C₂×#432×C₁₅₇ #34097276

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄²×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	#432×C₁₅₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₈×C₁₅₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₅₇	C₂	1	2	yes	maximal
5	C₂×#97×C₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄×C₁₅₇	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₅₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₅₇	C₂²	1	4	yes	—
10	C₂×C₄²×C₁₅₇	C₄	1	1	yes	characteristic
11	C₂×C₄²×C₁₅₇	C₄	1	4	yes	—
12	C₂×C₄²×C₁₅₇	C₄	1	1	yes	characteristic
13	C₂×C₄²×C₁₅₇	C₂²	1	4	yes	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₅₇	C₂²	1	8	yes	complemented
17	C₂²⋊C₈×C₁₅₇	C₂²	1	8	yes	—
18	#97×C₁₅₇	C₂²	1	8	yes	complemented
19	C₂⁴×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	—	2	4	no	—
21	C₂²×C₄×C₁₅₇	—	2	4	no	—
22	C₂²×C₄×C₁₅₇	—	2	4	no	—
23	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₁₅₇	—	2	8	no	—
34	C₂×C₈×C₁₅₇	C₂³	1	8	yes	—
35	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
36	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
37	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
38	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
39	C₄²×C₁₅₇	C₂³	1	2	yes	—
40	C₄²×C₁₅₇	C₂³	1	2	yes	—
41	C₂³×C₁₅₇	—	2	2	no	—
42	C₂³×C₁₅₇	—	2	2	no	—
43	C₂³×C₁₅₇	M₁₆	1	4	yes	—
44	C₂³×C₁₅₇	C₂²×C₄	1	2	yes	—
45	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	—	2	2	no	—
47	C₂×C₄×C₁₅₇	—	2	8	no	—
48	C₂×C₄×C₁₅₇	—	2	2	no	—
49	C₂×C₄×C₁₅₇	—	2	2	no	—
50	C₂×C₄×C₁₅₇	—	2	2	no	—
51	C₂×C₄×C₁₅₇	—	2	4	no	—
52	C₂×C₄×C₁₅₇	—	2	8	no	—
53	C₂×C₄×C₁₅₇	—	2	4	no	—
54	C₂×C₄×C₁₅₇	M₁₆	1	4	yes	—
55	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
59	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₅₇	#41	1	4	yes	—
62	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₅₇	—	2	8	no	—
64	C₂²×C₁₅₇	—	2	2	no	—
65	C₂²×C₁₅₇	—	2	2	no	—
66	C₂²×C₁₅₇	—	2	8	no	—
67	C₂²×C₁₅₇	#117	1	2	yes	—
68	C₂²×C₁₅₇	C₂×M₁₆	1	4	yes	—
69	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
70	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
71	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
72	C₄×C₁₅₇	—	2	2	no	—
73	C₄×C₁₅₇	—	2	2	no	—
74	C₄×C₁₅₇	—	2	2	no	—
75	C₄×C₁₅₇	—	2	2	no	—
76	C₄×C₁₅₇	#117	1	2	yes	—
77	C₄×C₁₅₇	C₂×M₁₆	1	4	yes	—
78	C₄×C₁₅₇	C₂×#41	1	2	yes	—
79	C₂×C₁₅₇	—	2	4	no	—
80	C₂×C₁₅₇	#432	1	4	yes	direct product factor
81	C₂×C₁₅₇	C₂×#117	1	1	yes	characteristic
82	C₂×C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic
83	C₂×C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic
84	C₁₅₇	C₂×#432	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×#432	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₄²	C₂×C₁₅₇	1	1	yes	characteristic
87	#432	C₂×C₁₅₇	1	8	yes	direct product factor
88	C₂×C₄⋊C₈	C₂×C₁₅₇	1	2	yes	complemented
89	C₂×C₂²⋊C₈	C₂×C₁₅₇	1	2	yes	—
90	C₂×#97	C₂×C₁₅₇	1	2	yes	complemented
91	C₂³×C₄	C₄×C₁₅₇	1	1	yes	characteristic
92	C₂³×C₄	C₄×C₁₅₇	1	1	yes	characteristic
93	C₂³×C₄	C₂²×C₁₅₇	1	1	yes	characteristic
94	C₂²×C₈	C₂²×C₁₅₇	1	4	yes	—
95	C₂×C₄²	C₄×C₁₅₇	1	1	yes	characteristic
96	C₂×C₄²	C₄×C₁₅₇	1	4	yes	—
97	C₂×C₄²	C₄×C₁₅₇	1	1	yes	characteristic
98	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
99	C₂×C₄²	C₂²×C₁₅₇	1	1	yes	characteristic
100	C₂×C₄²	C₂²×C₁₅₇	1	4	yes	—
101	C₄⋊C₈	C₂²×C₁₅₇	1	8	yes	complemented
102	C₂²⋊C₈	C₂²×C₁₅₇	1	8	yes	—
103	#97	C₂²×C₁₅₇	1	8	yes	complemented
104	C₂⁴	C₂×C₄×C₁₅₇	1	1	yes	characteristic
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂×C₄×C₁₅₇	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄×C₁₅₇	1	2	yes	—
110	C₂²×C₄	C₂×C₄×C₁₅₇	1	4	yes	—
111	C₂²×C₄	C₂×C₄×C₁₅₇	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×C₁₅₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₄×C₁₅₇	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×C₁₅₇	1	4	yes	—
115	C₂²×C₄	C₂×C₄×C₁₅₇	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₁₅₇	1	2	yes	—
117	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
118	C₂×C₈	—	2	8	no	—
119	C₂×C₈	C₂³×C₁₅₇	1	8	yes	—
120	C₄²	C₂×C₄×C₁₅₇	1	4	yes	—
121	C₄²	C₂×C₄×C₁₅₇	1	4	yes	—
122	C₄²	C₂×C₄×C₁₅₇	1	2	yes	—
123	C₄²	C₂×C₄×C₁₅₇	1	2	yes	—
124	C₄²	C₂³×C₁₅₇	1	2	yes	—
125	C₄²	C₂³×C₁₅₇	1	2	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	2	no	—
128	C₂³	M₁₆×C₁₅₇	1	4	yes	—
129	C₂³	C₂²×C₄×C₁₅₇	1	2	yes	—
130	C₂³	C₂²×C₄×C₁₅₇	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	8	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	M₁₆×C₁₅₇	1	4	yes	—
140	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
141	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
142	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
143	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
144	C₂×C₄	C₂²×C₄×C₁₅₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₄×C₁₅₇	1	2	yes	—
146	C₂×C₄	#41×C₁₅₇	1	4	yes	—
147	C₂×C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
148	C₈	—	2	8	no	—
149	C₂²	—	2	8	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	#117×C₁₅₇	1	2	yes	—
153	C₂²	C₂×M₁₆×C₁₅₇	1	4	yes	—
154	C₂²	C₂×M₁₆×C₁₅₇	1	2	yes	—
155	C₂²	C₂×M₁₆×C₁₅₇	1	2	yes	—
156	C₂²	C₂³×C₄×C₁₅₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	#117×C₁₅₇	1	2	yes	—
162	C₄	C₂×M₁₆×C₁₅₇	1	4	yes	—
163	C₄	C₂×#41×C₁₅₇	1	2	yes	—
164	C₂	—	2	4	no	—
165	C₂	#432×C₁₅₇	1	4	yes	direct product factor
166	C₂	C₂×#117×C₁₅₇	1	1	yes	characteristic
167	C₂	C₂²×M₁₆×C₁₅₇	1	1	yes	characteristic
168	C₂	C₂²×M₁₆×C₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₅₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₅₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₇
socle	C₂³×C₁₅₇	C₂²×C₄
prime radical	C₂⁴×C₁₅₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₅₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #432×C₁₅₇	C₂	direct (N × H)
3. C₂×C₄⋊C₈×C₁₅₇	C₂	semidirect
5. C₂×#97×C₁₅₇	C₂	semidirect
16. C₄⋊C₈×C₁₅₇	C₂²	semidirect
18. #97×C₁₅₇	C₂²	semidirect
80. C₂×C₁₅₇	#432	direct (N × H)
84. C₁₅₇	C₂×#432	direct (N × H)
85. C₂×#432	C₁₅₇	direct (N × H)
87. #432	C₂×C₁₅₇	direct (N × H)
88. C₂×C₄⋊C₈	C₂×C₁₅₇	semidirect
90. C₂×#97	C₂×C₁₅₇	semidirect
101. C₄⋊C₈	C₂²×C₁₅₇	semidirect
103. #97	C₂²×C₁₅₇	semidirect
165. C₂	#432×C₁₅₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄²×C₁₅₇	2
#432×C₁₅₇	2
C₂×C₄⋊C₈×C₁₅₇	2
C₂×C₂²⋊C₈×C₁₅₇	2
C₂×#97×C₁₅₇	2
C₂×#432	157

Subgroups — C2×#432×C157 #34097276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#432×C₁₅₇ #34097276